Chapter 13
The Accuracy of Averages
13.1
Estimators and estimates
As its name suggests, the aim of estimation is to determine the approximate
value of the population parameter on the basis of a sample statistic, which is
the essence of statistical inference. It is important to differentiate however
between an
estimator
and an
estimate
. Roughly, an estimator is a rule of
finding an estimate of a population parameter. In other words, we find an
estimate of a population parameter by employing a certain estimator.
In statistics, it is desirable to use a good estimator.
Two important
criteria for making some judgment of estimators are 1) unbiasedness and
2) efficiency or minimum variance.
An estimator is said to be unbiased if the average value of all the es
timates is equal to the population parameter being estimated. For exam
ple, the arithmetic average
X
=
1
n
∑
X
i
is an unbiased estimator of the
population mean,
μ
.
In contrast, the standard deviation calculated by
1
n
∑
(
X

X
)
2
is a biased estimator of the population standard deviation.
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 Spring '10
 YY
 population parameter, draws

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